In modern distributed storage systems, node failures are common and single-node failures
occur most frequently among all failures [1]}, [2]}. It is important to
repair the failed node with the repair bandwidth defined as the total amount of
symbols downloaded from other surviving nodes as small as possible [3]}.
Recently many constructions of \((n,k,m)\)  MDS array codes to achieve the minimum repair bandwidth
\(\frac{dm}{d-k+1}\)  from \(d\)  surviving nodes have been proposed in [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, all the high-code-rate
(i.e.,\(k/n > 0.5\) ) MDS array codes with the minimum repair bandwidth need an
exponential sub-packetization level in parameters \(n\)  and \(k\)  [10]}.
It is practical important to design high-code-rate MDS array codes with repair bandwidth
as small as possible, for a given small sub-packetization level. HashTag Erasure Codes (HTEC)
[11]} is a high-code-rate that have efficient repair method for data nodes with
sub-packetization level \(2 \le m \le r^{\lceil \frac{k}{r}\rceil }\) , however no efficient
repair method for parity nodes and the required field size
should be large enough to keep the MDS property.
